acta mechanica et automatica, vol.8 no.3 (2014), DOI 10.2478/ama-2014-0022
TETRAGONAL OR HEXAGONAL SYMMETRY IN MODELING OF FAILURE CRITERIA
FOR TRANSVERSELY ISOTROPIC MATERIALS
Artur GANCZARSKI*, Michał ADAMSKI*
*Institute of Applied Mechanics, Department Mechanical Engineering,
Cracow University of Technology, 31-864 Kraków al. Jana Pawła II 37, Poland
[email protected], [email protected]
Abstract: Present work deals with modeling of failure criteria for transversely isotropic materials. Analysis comprises two classes of symmetry: Tsai-Wu tetragonal and new Tsai-Wu based hexagonal. Detail analysis of both classes of symmetry with respect to their advantages as well as limitations is presented. Finally, simple comparison of differences between limit curves corresponding to cross sections by planes of transverse isotropy, orthotropy and shear plane is done.
Key words: Transverse Isotropy, Tetragonal and Hexagonal Symmetry, Tsai-Wu Criterion
1. INTRODUCTION
Formulation of the initial yield or failure criteria for modern materials has to consider material anisotropy, tension-compression
asymmetry or hydrostatic pressure sensitivity. In general, there
exist two competitive but complementary schools to capture the
above behaviours. The first explicit approach is based on a direct
concept of common invariants of stress and anisotropy tensors
(Sayir, 1970; Goldenblat and Kopnov, 1966; Tsai and Wu, 1971;
Murakami, 2012; etc.) The second implicit approach is based
on a direct extension of the isotropic-type yield/failure criteria
to capture anisotropic behaviour as well as strength differential
effect and pressure sensitivity by the use of linear transformation
tensors for single stress invariants (Khan et al., 2007; Cazacu and
Barlat, 2004 and others). Although the common invariants-based
formulation is more mathematically rigorous, but complicated by
the use of the second-, fourth-rank structural tensors, a direct
extension of classical isotropic criteria to anisotropy is very efficient and broadly examined (Ganczarski and Skrzypek, 2014).
Basic result of the present paper is to propose the new orthotropic criterion of failure initiation, being the extended Tsai-Wu
type equation. This new criterion is capable of capturing orthotropic limit surface description, without ellipticity loss even
for arbitrarily high orthotropy degree. This is by contrast to the
classical Tsai-Wu (1971) criterion (eg. Ottosen and Ristinmaa,
2005), where for high orthotropy degree inadmissible degeneration of the limit surface may occur. This enhanced Tsai-Wu's type
limit surface is represented by the elliptic paraboloid, the axis
of which is different from hydrostatic axis in the space of principal
stresses, hence the criterion no longer satisfies the property
of deviatoricity.
The other important feature of the proposed criterion is concerned with the particular case of transversely isotropic hexagonal
symmetry, for which it is possible to achieve coincidence with the
same proportion of isotropic Huber-von Mises-Hencky condition
in the transverse isotropy plane, which may occur beneficial when
compared to the deviatoric Tsai-Wu formulation for which above
reducibility does not hold.
2. TRANSVERSELY ISOTROPIC TSAI-WU FAILURE
CRITERION
In a general case of brittle materials that exhibit anisotropy
(e.g. concrete, ceramic materials, rocks, composite materials,
etc.) and tension-compression asymmetry, a transversely isotropic
Tsai-Wu criterion of initial failure (Tsai and Wu, 1971) is applicable:
2
𝐹[(𝜎𝑦 − 𝜎𝑧 )2 + (𝜎𝑧 − 𝜎𝑥 )2 ] + 𝐻(𝜎𝑥 − 𝜎𝑦 )2 + 2𝐿𝜏𝑥𝑦
2
2
+2𝑀(𝜏𝑧𝑥 + 𝜏𝑧𝑦 ) + 𝑃(𝜎𝑥 + 𝜎𝑦 ) + 𝑄𝜎𝑧 = 1
(1)
Eq. (1) contains only 5 independent material coefficients referring to appropriate tensile and compressive strengths k tx , k cx ,
k tz , k cz and shear strength k zx , hence, in order to calibrate it the
following tests must be performed:
𝜎𝑥 = 𝑘𝑡𝑥
⟶
2
(𝐹 + 𝐻)𝑘𝑡𝑥
+ 𝑃𝑘𝑡𝑥 = 1
𝜎𝑥 = −𝑘𝑐𝑥
⟶
2
(𝐹 + 𝐻)𝑘𝑐𝑥
− 𝑃𝑘𝑐𝑥 = 1
𝜎𝑧 = 𝑘𝑡𝑧
⟶
2
2𝐹𝑘𝑡𝑧
+ 𝑄𝑘𝑡𝑧 = 1
𝜎𝑧 = −𝑘𝑐𝑧
⟶
2
2𝐹𝑘𝑐𝑥
𝜏𝑧𝑥 = 𝑘𝑧𝑥
⟶
2
2𝑀𝑘𝑧𝑥
=1
(2)
− 𝑄𝑘𝑐𝑧 = 1
Solution of Eqs. (2) with respect to F, H, M, P and Q takes the
form:
𝐹=
𝑃=
1
2𝑘𝑡𝑧 𝑘𝑐𝑧
1
1
𝑘𝑡𝑥
−
𝑘𝑐𝑥
𝐻=
1
−
𝑘𝑡𝑥 𝑘𝑐𝑥
1
𝑄=
𝑘𝑡𝑧
−
1
2𝑘𝑡𝑧 𝑘𝑐𝑧
1
𝑀=
1
2
2𝑘𝑧𝑥
(3)
𝑘𝑐𝑧
Magnitude of material coefficient 𝐿, referring to shear strength
in plane of transverse isotropy, is not independent and can be
calculated from the following known relation (see Chen and Han,
1995 and Ganczarski and Skrzypek, 2013):
2𝐿 = 2(𝐹 + 2𝐻) =
4
𝑘𝑡𝑥 𝑘𝑐𝑥
−
1
𝑘𝑡𝑧 𝑘𝑐𝑧
(4)
Hence, after substitution of Eqs. (3)–(4) to Eq. (1) one can get
125
Artur Ganczarski, Michał Adamski
Tetragonal or Hexagonal Symmetry in Modeling of Yield Criteria for Transversely Isotropic Materials
final form of the transversely isotropic Tsai-Wu criterion:
𝜎𝑥2 +𝜎𝑦2
𝑘𝑡𝑥 𝑘𝑐𝑥
+
𝜎𝑧2
𝑘𝑡𝑧 𝑘𝑐𝑧
+(
+(
1
𝑘𝑡𝑥
−
2
−(
𝑘𝑡𝑥 𝑘𝑐𝑥
4
𝑘𝑡𝑥 𝑘𝑐𝑥
1
𝑘𝑐𝑥
−
1
−
1
𝑘𝑡𝑧 𝑘𝑐𝑧
𝑘𝑡𝑧 𝑘𝑐𝑧
)𝜎𝑥 𝜎𝑦 −
2
)𝜏𝑥𝑦
+
)(𝜎𝑥 +𝜎𝑦 ) + (
1
𝑘𝑡𝑧
2 +𝜏2
𝜏𝑦𝑧
𝑧𝑥
−
𝑘𝑡𝑧 𝑘𝑐𝑧
2
𝑘𝑧𝑥
1
𝑘𝑐𝑧
a
𝜎𝑦 𝜎𝑧 +𝜎𝑥 𝜎𝑧
(5)
)𝜎𝑧 = 1
It is obvious that material coefficients in plane of transverse
isotropy that precede terms σx σy and τxy are not fully independent since they contain not only in-plane tensile and compressive
strengths k tx , k cx but also out-of-transverse isotropy plane tensile
and compressive strengths k tz , k cz . Consequently, Eq. (5) can be
classified as the tetragonal transversely isotropic Tsai-Wu criterion
of initial failure.
3. CONVEXITY LOSS IN CASE OF HIGH ORTHOTROPY
Applicability range of Tsai-Wu transversely isotropic criterion
(5) to properly describe initiation of failure in some engineering
materials that exhibit high orthotropy degree, is bounded by
a possible elliptic form loss of the limit surface. In other words,
a physically inadmissible degeneration of the single convex
and simply connected elliptic limit surface into two concave hyperbolas surfaces occurs. The following inequality bounds the
range of applicability for Hill's criterion (see Ottosen and Ristinmaa, 2005; Ganczarski and Skrzypek, 2013):
1
4
(
𝑘𝑡𝑧 𝑘𝑐𝑧 𝑘𝑡𝑥 𝑘𝑐𝑥
−
1
𝑘𝑡𝑧 𝑘𝑐𝑧
)>0
b
(6)
Substitution of the dimensionless parameter R =
2(k tz k cz /k tx k cx ) − 1 (extension of Hosford and Backofen
(1964) parameter), leads to the simplified restriction:
𝑅 > −0.5
(7)
If the above inequalities (6)–(7) do not hold, elliptic cross sections of the limit surface degenerate to two hyperbolic branches
and the loss of convexity occurs. To illustrate this limitation, the
failure curves in two planes: the transverse isotropy (σx , σy ):
𝜎𝑥2 −
2𝑅
1+𝑅
𝜎𝑥 𝜎𝑦 + 𝜎𝑦2 + (𝑘𝑐𝑥 − 𝑘𝑡𝑥 )(𝜎𝑥 +𝜎𝑦 ) = 𝑘𝑡𝑥 𝑘𝑐𝑥 (8)
and the orthotropy plane (σx , σz ):
𝜎𝑥2 −
2
1+𝑅
𝜎𝑥 𝜎𝑧 +
+𝑘𝑡𝑥 𝑘𝑐𝑥 (
1
2
1+𝑅
𝑘𝑡𝑧
−
𝜎𝑧2 + (𝑘𝑐𝑥 − 𝑘𝑡𝑥 )𝜎𝑥
1
𝑘𝑐𝑧
)𝜎𝑧 = 𝑘𝑡𝑥 𝑘𝑐𝑥
(9)
for various R–values, are sketched in Fig. 1a, b respectively. It is
observed that when R, starting from R = 3, approaches the limit
R = −0.5, the curves change from closed ellipses to two parallel
lines in transverse isotropy plane or parabola in orthotropy plane,
whereas for R < −0.5, concave hyperbolas appear. In general
case of strong orthotropy, when the convexity condition (6) does
not hold, the Tsai-Wu criterion (5) becomes useless. This effect is
decribed in details for the case of Hill’s (1948) orthotropic yield
criterion (see Ganczarski and Skrzypek, 2013) from which TsaiWu’s criterion obviously inherits all inconvenient features since
the structure of quadratic terms of both conditions is analogous.
126
Fig. 1. Degeneration of the Tsai-Wu limit surface with the magnitude
of the generalized Hosford and Backhofen parameter 𝑅:
a) transverse isotropy plane, b) orthotropy plane
4. MODIFIED TSAI–WU BASED HEXAGONAL FAILURE
CRITERION
Except the tetragonal transversely isotropic Tsai-Wu criterion
Eq. (5) one can introduce hexagonally isotropic Tsai-Wu failure
criterion:
𝜎𝑥2 −𝜎𝑥 𝜎𝑦 +𝜎𝑦2
𝑘𝑡𝑥 𝑘𝑐𝑥
2 +𝜏2
𝜏𝑦𝑧
𝑧𝑥
2
𝑘𝑧𝑥
+(
+
1
𝑘𝑡𝑥
𝜎𝑧2
𝑘𝑡𝑧 𝑘𝑐𝑧
−
1
𝑘𝑐𝑥
−
𝜎𝑦 𝜎𝑧 +𝜎𝑥 𝜎𝑧
𝑘𝑡𝑧 𝑘𝑐𝑧
)(𝜎𝑥 +𝜎𝑦 ) + (
+
3
𝜏2
𝑘𝑡𝑥 𝑘𝑐𝑥 𝑥𝑦
1
−
𝑘𝑡𝑧
1
𝑘𝑐𝑧
+
)𝜎𝑧 = 1
(10)
in which coefficients that precede terms σx σy and τ2xy are always
positive. These prevent elliptic form of failure curves from degeneration and reduce Eq. (10) to the Huber-von Mises-Hencky ellipse “shifted” outside the origin of co-ordinate system in case
of transverse isotropy plane. Since the Huber-von Mises-Hencky
acta mechanica et automatica, vol.8 no.3 (2014), DOI 10.2478/ama-2014-0022
criterion exhibits isotropy in hexagonal sense, therefore the TsaiWu failure criterion can also be classified as hexagonal in the
plane of transverse isotropy. Consequently, condition (10) never
violates the Drucker stability postulate, which is not guaranteed by
equation (5). However, the hexagonally isotropic Tsai-Wu failure
criterion cannot be presented in explicitly deviatoric form (1) which
means that the axis of appropriate limit surface in the principal
stress space is not parallel to the hydrostatic axis.
er than tensile strength k tz , whereas analogous ratio k cx /k tx
is approximately equal to 7 in case of transverse isotropy plane.
Moreover, ratio of semi-axes for Tsai-Wu tetragonal ellipse
in (σx , σy ) plane essentially exceeds analogous ratio for Hubervon Mises-Hencky like ellipse, contrary to the case of Tsai–Wu
hexagonal ellipse.
5. RESULTS
Both the Tsai-Wu transversely isotropic initial failure criteria:
tetragonal Eq. (5) and new hexagonal type Eq. (10) are compared
for columnar ice in plane of transverse isotropy (σx , σy ) in Fig. 2,
in plane of orthotropy (σx , σz ) in Fig. 3 and in plane of transverse
isotropy (σx , τxy ) in Fig. 4. The experimental data of columnar
ice was established by Ralston (1997) in Tab. 1.
Tab. 1. Experimental data for columnar ice, after Ralson (1997)
Tensile strength
ktx
1.01 MPa
ktz
1.21 MPa
Compressive strength
kcx
7.11 MPa
kcz
13.5 MPa
Cross sections of the limit surface are ellipses, that exhibit
strong aspect ratio in case tetragonal symmetry, the centers
of which are shifted outside the origin of co-ordinate system towards the quarter referring to compressive stresses. In case
of cross section by plane of transverse isotropy (see Fig. 2) the
symmetry axis has obviously inclination equal 45° to the axes
of coordinate system, in other words it overlaps projection
of hydrostatic axis at the transverse isotropy plane (σx , σy ), contrary to the cross section by plane of orthotropy (see Fig. 3) the
main semi-axis of ellipse is inclined by 71.1°.
Fig. 3. Comparison of transversely isotropic Tsai-Wu initial failure criteria
of tetragonal and hexagonal types for columnar ice
in case of plane of orthotropy (𝜎𝑥 , 𝜎𝑧 )
In case of cross sections of the limit surface in shear plane
(σx , τxy ) both ellipses referring to tetragonal and to hexagonal
symmetry have comparable aspect ratio along tensioncompression direction mainly resulting from aforementioned
strength difference ratio k cx /k tx however the criterion (10) performs ellipse slightly broder than the criterion (5).
Fig. 4. Transverse isotropic Tsai-Wu initial failure criteria of tetragonal
and hexagonal types for columnar ice
in case of plane of transverse isotropy (𝜎𝑥 , 𝜏𝑥𝑦 )
Fig. 2. Comparison of transversely isotropic Tsai-Wu initial failure criteria
of tetragonal and hexagonal types for columnar ice
in case of plane of transverse isotropy (𝜎𝑥 , 𝜎𝑦 )
It has to be emphasized that, in case of columnar ice, compressive strength along othotropy axis k cz is over 10 times great-
It is also worth to emphasize that although the tetragonal
transversely isotropic Tsai-Wu failure criterion Eq. (5) and the
hexagonal transversely isotropic Tsai-Wu failure criterion Eq. (10)
contain the same number of 5 independent strengths k tx , k cx ,
k tz , k cz and k zx , only criterion (10) is free from convexity loss
and simultaneously truly transversely isotropic in sense of hexagonal class of symmetry.
127
Artur Ganczarski, Michał Adamski
Tetragonal or Hexagonal Symmetry in Modeling of Yield Criteria for Transversely Isotropic Materials
6. CONCLUSIONS
Both transversely isotropic failure criteria: Tsai-Wu tetragonal
and Tsai-Wu hexagonal perform paraboloidal surfaces in the
space of principal stresses. However, the tetragonal criterion
is represented by elliptic paraboloid which axis is parallel to the
hydrostatic axis contrary to the hexagonal one, which represents
elliptic paraboloid which axis is not parallel to the hydrostatic axis.
Hence, hexagonal Tsai-Wu failure criterion does not satisfy deviatoric property; this is a consequence of its coincidence with Hubervon Mises-Hencky like criterion in the plane of transverse isotropy
as well as a property of saving elliptical nature despite of high
ratio of orthotropy. Choice of appropriate transversely yield criterion either tetragonal or hexagonal depends on coincidence with
experimental tests done on real materials, that can be subjected
to one or other class of symmetry, but also can exhibit properties
different than aforementioned cases. Key point for proper classification of real transversely isotropic material to one of symmetry
class (tetragonal, hexagonal) is the shape of limit curve belonging
to the plane of transverse isotropy.
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The work was supported by National Science Centre Poland grant
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